Dependencies

The abscissa of absolute convergence of \(L(\text{pow}^R(\nu ), s)\) is at most \(\Re (\nu ) + 1\).

If \(A,B,C,R {\gt} 0\) then the classical bound is monotone decreasing for \(x \geq \exp ( R (2B/C)^2 )\).

An application of the Maximum modulus principle.

Let \(\chi : G \to \mathbb {C}^\times \) be a nontrivial character. Then \(L(\chi ,s)\) extends to an analytic function on the half-plane \(\Re (s) {\gt} 1 - [K:\mathbb {Q}]^{-1}\), and in particular is holomorphic at \(s=1\).

Let \(\chi : G \to \mathbb {C}^\times \) be an abelian character. For \(\Re (s) {\gt} 1\) one has

where the product runs over nonzero prime ideals of \(\mathcal{O}_K\), with the convention \(\chi (\mathfrak {p}) = 0\) if \(\mathfrak {p}\) ramifies in the fixed field of \(\ker \chi \).

One has

for all \(x \geq 1\).

The balance of a factorization \(a_1 \dots a_t\) at a prime \(p\) is defined as the number of times \(p\) divides \(a_1 \dots a_t\), minus the number of times \(p\) divides \(n!\).

If a factorization has zero total imbalance, then it exactly factors \(n!\).

If \(f\) is bounded above in a punctured neighborhood of \(p\), then \(f\) is \(O(1)\) in that neighborhood.

Let \(g : \mathbb {C}\to \mathbb {C}\) be a holomorphic function on a rectangle, then \(g\) is bounded above on the rectangle.

The Fourier transform is a bijection on the Schwartz class. [Note: only surjectivity is actually used.]

If \(f:\mathbb {C}\to \mathbb {C}\) is analytic on \(\overline{\mathbb {D}_R}\) then so too is \(B_f\).

Let \(0 {\lt} r {\lt} R {\lt} 1\), and \(f:\mathbb {C}\to \mathbb {C}\) be analytic \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). We define a function \(B_f:\mathbb {C}\to \mathbb {C}\) as follows.

Let \(0 {\lt} r {\lt} R{\lt}1\) and \(f:\overline{\mathbb {D}_1}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). Then \(B_f(z)\neq 0\) for all \(z\in \overline{\mathbb {D}_r}\).

Let \(0 {\lt} r {\lt} R{\lt}1\), and \(f:\mathbb {C}\to \mathbb {C}\) be analytic on \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). Then

An application of

      Complex.borelCaratheodory_zero.
    

Let \(R,\, M{\gt}0\). Let \(f\) be analytic on \(|z|\leq R\) such that \(f(0)=0\) and suppose \(\Re f(z)\leq M\) for all \(|z|\leq R\). Then for any \(0 {\lt} r {\lt} R\),

for all \(|z|\leq r\).

If \(M\) is sufficiently large depending on \(\varepsilon \), then \(n \log (1-1/M)^{-1} \leq \varepsilon n\).

If \(L\) is sufficiently large depending on \(M, \varepsilon \), and \(n\) sufficiently large depending on \(L\), then \(\sum _{p \leq n/L} M \log n \leq \varepsilon n\).

If \(n\) sufficiently large depending on \(M, \varepsilon \), then \(\sum _{p \leq \sqrt{n}} M \log ^2 n / \log 2 \leq \varepsilon n\).

If \(n\) sufficiently large depending on \(L, \varepsilon \), then \(\sum _{n/L {\lt} p \leq n} \frac{n}{p} \log \frac{n}{p} \leq \varepsilon n\).

If \(n\) sufficiently large depending on \(M, L, \varepsilon \), then \(\sum _{p \leq L} (M \log n + M L \pi (n)) \log L \leq \varepsilon n\).

Starting from any factorization \(f\), one can find a factorization \(f'\) in balance whose cardinality is at most \(\log n!\) plus the score of \(f\), divided by \(\log n\).

Let \(f\) be analytic on \(|z|\leq R\). For any \(z\) with \(|z|\leq r\) and any \(r'\) with \(0 {\lt} r {\lt} r' {\lt} R\) we have

If \(f:\mathbb {C}\to \mathbb {C}\) is analytic on \(\overline{\mathbb {D}_1}\) then so too is \(C_f\).

Let \(0 {\lt} r {\lt} 1\), and \(f:\mathbb {C}\to \mathbb {C}\) be analytic on \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). We define a function \(C_f:\mathbb {C}\to \mathbb {C}\) as follows. This function is constructed by dividing \(f(z)\) by a polynomial whose roots are the zeros of \(f\) inside \(\overline{\mathbb {D}_r}\).

where \(h_z(z)\) comes from Lemma 5.3.4.

Let \(L/K\) be finite abelian with Galois group \(G\). For every \(\sigma \in G\), the Dirichlet density of primes of \(K\) with Frobenius \(\sigma \) exists and equals \(1/|G|\).

Choose an integer \(m\geq 1\) not dividing the discriminant of \(L/K\), large enough that \(H := \mathrm{Gal}(L(\mu _m)/L) \cong (\mathbb {Z}/m\mathbb {Z})^\times \) via the cyclotomic character and \(\mathrm{Gal}(L(\mu _m)/K) \cong G \times H\). For \(\sigma \in G\) and \(\tau \in H\) write \(S_{\sigma ,\tau }\) for the set of primes of \(K\) unramified in \(L(\mu _m)\) with Frobenius \((\sigma ,\tau )\in G\times H\), and \(S_\sigma \) for the set of primes of \(K\) unramified in \(L\) with Frobenius \(\sigma \) in \(G\). Then

In the notation of Lemma 3.6.1, if the order of \(\tau \in H\) is divisible by \(|G|\), then \(\langle (\sigma ,\tau )\rangle \cap (G\times \{ 1\} ) = \{ 1\} \), so \(L(\mu _m)\) is obtained by adjoining \(\mu _m\) to the fixed field \(F := L(\mu _m)^{\langle (\sigma ,\tau )\rangle }\). Applying Proposition 3.5.1 to the cyclotomic extension \(F(\mu _m)/F\) and transporting densities as in the reduction step of Theorem 7.2.2 yields \(\delta (S_{\sigma ,\tau }) = 1/(|G|\, |H|)\).

Let \(m\geq 1\) and \(G = \mathrm{Gal}(\mathbb {Q}(\mu _m)/\mathbb {Q}) \cong (\mathbb {Z}/m\mathbb {Z})^\times \). For any \(a\) coprime to \(m\),

In particular the primes in the progression \(a\pmod{m}\) have natural density \(1/\varphi (m)\) (Dirichlet’s theorem).

For every \(\sigma \in G\), the set of prime ideals \(\mathfrak {p}\) of \(K\) unramified in \(L\) with Frobenius \(\varphi _{\mathfrak {p}} = \sigma \) has Dirichlet density \(1/|G|\).

Let \(L/K\) be a finite Galois extension of number fields with Galois group \(G\), and let \(C\subset G\) be a conjugacy class. The set of prime ideals \(\mathfrak {p}\) of \(K\) unramified in \(L\) whose Frobenius conjugacy class equals \(C\) has Dirichlet density \(|C|/|G|\).

Fix \(\sigma \in G\). For every prime ideal \(\mathfrak {p}\) of \(\mathcal{O}_K\) unramified in \(L\),

Equivalently, if \(\sigma (\zeta _m) = \zeta _m^a\) with \(\gcd (a,m)=1\), the sum equals \(|G|\) precisely when \(N\mathfrak {p} \equiv a \pmod{m}\).

Fix \(\sigma \in C\) and let \(E\) be the fixed field of \(\langle \sigma \rangle \), so \(L/E\) is cyclic of degree \(f = |\langle \sigma \rangle |\). Write \(S\) for the set of primes of \(K\) unramified in \(L\) whose Frobenius class equals \(C\), and \(T_\sigma \) for the set of primes \(\mathfrak {P}\) of \(E\) unramified in \(L\) (and lying over \(K\)) with Frobenius equal to \(\sigma \). Then

whenever either density exists. In particular, the Chebotarev statement for \(L/K\) and \(C\) follows from the abelian statement for the cyclic extension \(L/E\) and the element \(\sigma \).

One has

If \(a\ (q)\) is a primitive residue class, then one has

The (second) Chebyshev Psi function is defined as

where \(\Lambda (n)\) is the von Mangoldt function.

We say that \(E_\pi \) satisfies a classical bound with parameters \(A, B, C, R, x_0\) if for all \(x \geq x_0\) we have

We say that it obeys a numerical bound with parameter \(\varepsilon (x_0)\) if for all \(x \geq x_0\) we have

We say that \(E_\psi \) satisfies a classical bound with parameters \(A, B, C, R, x_0\) if for all \(x \geq x_0\) we have

We say that it obeys a numerical bound with parameter \(\varepsilon (x_0)\) if for all \(x \geq x_0\) we have

We say that \(E_\theta \) satisfies a classical bound with parameters \(A, B, C, R, x_0\) if for all \(x \geq x_0\) we have

We say that it obeys a numerical bound with parameter \(\varepsilon (x_0)\) if for all \(x \geq x_0\) we have

A classical bound for \(x \geq x_0\) implies a numerical bound for \(x \geq \max (x_0, \exp ( R (2B/C)^2 ))\).

For \(1{\lt}\Re (s)\) and \(p\) prime, we have that \(1+p^{-s}\neq 0\). The naming convention is to mimic

      Complex.one_sub_prime_cpow_ne_zero
    

For \(x{\gt}1\) and \(\sigma {\lt}-3/2\), we have

If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported with \(f\) and \(\hat\psi \) non-negative, then there exists a constant \(B\) such that

for all \(x {\gt} 0\).

\(d_k\) is the \(k\)-fold divisor function: the number of ways to write \(n\) as an ordered product of \(k\) natural numbers. Equivalently, the Dirichlet convolution of \(\zeta \) with itself \(k\) times.

(1.25) in Iwaniec-Kowalski: a formula for \(d_k\) for all \(n\).

Explicit formula: \(d_k (p^a) = (a + k - 1).choose (k - 1)\) for prime \(p\) and \(k \geq 1\).

\(d_k\) is multiplicative for all \(k\).

\(d_1\) is \(\zeta \).

Recurrence: \(d_{k+1} = d_k * \zeta \).

\(d_2\) is the classical divisor count function \(\tau \).

\(d_0\) is the multiplicative identity (indicator at 1).

If \(\psi :\mathbb {R}\to \mathbb {C}\) is \(C^2\) and obeys the bounds

for all \(t \in \mathbb {R}\), then

for all \(u \in \mathbb {R}\), where \(C\) is an absolute constant.

If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable, absolutely continuous, and \(\psi '\) is of bounded variation, then

for all \(u \in \mathbb {R}\).

For \(\Re (s) {\gt} 1\) one has

where the product runs over all (necessarily one-dimensional) irreducible characters of \(G\).

For any nontrivial character \(\chi \) of \(G = \mathrm{Gal}(L/K)\), one has \(L(\chi ,1) \neq 0\). Moreover \(L(\chi ,s)\) does not vanish for \(\Re (s)=1\).

For any nontrivial character \(\chi \) of \(G\),

(Equivalently, writing \(\Lambda _\chi \) for the von Mangoldt-type coefficients of \(-\frac{L'}{L}(\chi ,s)\), one has \(\sum _{n\leq x} \Lambda _\chi (n) = o(x)\).)

The Dedekind zeta function \(\zeta _L\) admits a meromorphic continuation to a neighbourhood of the line \(\Re (s)=1\) with a single simple pole at \(s=1\) (and is otherwise holomorphic and nonvanishing on that line after removing the pole). In particular \(\log \zeta _L(s) \sim \log (s-1)^{-1}\) as \(s \to 1^+\).

For all \(t\in \mathbb {R}\) with \(|t|\geq 2\) we have that

Let \(\nu \) be a bumpfunction supported in \([1/2,2]\). Then for any \(\epsilon {\gt}0\), we define the delta spike \(\nu _\epsilon \) to be the function from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) defined by

For any \(\epsilon {\gt}0\), we have

Let \(\delta _t=E/\log |t|\) where \(E\) is the constant coming from Theorem 5.3.21.

Let \(f : \mathbb {C} \to \mathbb {C}\) be a function at \(p \in \mathbb {C}\) with derivative \(a\). Then the derivative of the function \(g(z) = \overline{f(\overline{z})}\) at \(\overline{p}\) is \(\overline{a}\).

Conjugation symmetry of the derivative of the Riemann zeta function. Let \(s \in \mathbb {C}\). Then

Let \(R,\, M{\gt}0\) and \(0 {\lt} r {\lt} r' {\lt} R\). Let \(f\) be analytic on \(|z|\leq R\) such that \(f(0)=0\) and suppose \(\Re f(z)\leq M\) for all \(|z|\leq R\). Then we have that

for all \(|z|\leq r\).

For any \(s = \sigma + tI \in \mathbb {C}\), \(1/2 \le \sigma \le 2, 3 {\lt} |t|\), and any \(0 {\lt} A {\lt} 1\) sufficiently small, and \(1-A/\log |t| \le \sigma \), we have

Let \(x{\gt}0\). Then for \(0 {\lt} c {\lt} 1 /2\), we have that the function

is bounded above on the rectangle with corners at \(-1-c-i*c\) and \(-1+c+i*c\) (except at \(s=-1\)).

Let \(x{\gt}0\). Then for \(0 {\lt} c {\lt} 1 /2\), we have that the function

is bounded above on the rectangle with corners at \(-c-i*c\) and \(c+i*c\) (except at \(s=0\)).

The difference of two vertical integrals and a rectangle is the difference of an upper and a lower U integrals.

Any primitive residue class contains an infinite number of primes.

Let \(0 {\lt} r {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\) such that \(|f(z)|\leq B\) for \(|z|\leq R\), then \(|B_f(z)|\leq B\) for \(|z|\leq R\) also.

Let \(m\) and \(n\) be coprime natural numbers. The function \((a,b) \mapsto ab\) is injective on the product of the divisors of \(m\) and \(n\).

    Upstreamed to mathlib via PR #36495.
    

For \(\sigma _0 {\gt} 1\), there exists a constant \(C {\gt} 0\) such that

One has \(E_1 \leq \frac{5 \log 2 + 3}{4}\)

The series \(E_1 := \sum _p \frac{\log p}{p(p-1)}\) converges.

\(E_\pi (x) = |\pi (x) - \mathrm{Li}(x)| / (x / \log x)\).

\(E_\psi (x) = |\psi (x) - x| / x\)

One can find a balanced factorization of \(n!\) with cardinality at most \(n - n / \log n + o(n / \log n)\).-

One can factorize \(n!\) into at most \(n/2 - n / 2\log n + o(n / \log n)\) numbers of size at most \(n^2\).-

\(E_\theta (x) = |\theta (x) - x| / x\)

We set \(\gamma := \int _2^\infty \frac{E_{1,\Lambda }(t)}{t \log ^2 t} \, dt + 1 - \log \log 2\).

\(\gamma \) is the Euler–Mascheroni constant.

If \(f\) is differentiable on a set \(s\) except at \(c\in s\), and \(f\) is bounded above on \(s\setminus \{ c\} \), then there exists a differentiable function \(g\) on \(s\) such that \(f\) and \(g\) agree on \(s\setminus \{ c\} \).

We work with (approximate) factorizations \(a_1 \dots a_t\) of a factorial \(n!\).

Let \(B{\gt}1\) and \(0 {\lt} r' {\lt} r {\lt} R' {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)=1\) and \(|f(z)|\leq B\) for all \(|z|\leq R\), then for all \(z\in \overline{\mathbb {D}_{r'}}\setminus \mathcal{K}_f(R')\) we have

For any arithmetic function \(f\) and real number \(x\), one has

and

If \(\psi : \mathbb {R}\to \mathbb {C}\) is integrable and \(x {\gt} 0\), then for any \(\sigma {\gt}1\)

\(P(g)\) is the first prime \(p_n\) for which the prime gap \(p_{n+1}-p_n\) is equal to \(g\), or \(0\) if no such gap exists.

For \(x{\gt}1\) and \(\sigma {\gt}0\), we have

For \(x{\gt}0\), \(\sigma {\gt}0\), and \(x{\lt}1\), we have

For \(\sigma {\gt}1\) and \(t\in \mathbb {R}\), define

Then \(F\) is real-valued, and whence \(F(\sigma )\ge 1\) there.

Let \(t\in \mathbb {R}\) with \(|t|\geq 3\) and \(z=\sigma +it\) where \(1-\delta _t/3\leq \sigma \leq 3/2\). Additionally, let \(\rho \in \mathcal{Z}_t\). Then we have that

Assuming a bound of the form of Definition 4.5.3, we have that

For all \(s\in \mathbb {C}\) with \(|s|\leq 1\) and \(t\in \mathbb {R}\) with \(|t|\geq 2\), we have that

If \(f\) is holomorphic on a rectangle \(z\) and \(w\), then the integral of \(f\) over the rectangle with corners \(z\) and \(w\) is \(0\).

For any \(N\ge 1\), the function \(\zeta _0(N,s)\) is holomorphic on \(\{ s\in \mathbb {C}\mid \Re (s){\gt}0 ∧ s \ne 1\} \).

We have that

Assuming a bound of the form of Definition 4.5.3 we have that

Assuming a bound of the form of Definition 4.5.3, we have that

Assuming a bound of the form of Definition 4.5.3, we have that

We have that

Assuming a bound of the form of Definition 4.5.3, we have that

Assuming a bound of the form of Definition 4.5.3, we have that

Assuming a bound of the form of Definition 4.5.3, we have that

Symmetry between \(I_2\) and \(I_8\):

We have that

Given a natural number \(k\) and a real number \(X_0 {\gt} 0\), there exists \(C \geq 1\) so that for all \(X \geq X_0\),

The number of elements in this initial factorization is at most \(n\).

We perform an initial factorization by taking the natural numbers between \(n-n/M\) (inclusive) and \(n\) (exclusive) repeated \(M\) times, deleting those elements that are not \(n/L\)-smooth (i.e., have a prime factor greater than or equal to \(n/L\)).

A large prime \(p \geq n/L\) can be in deficit by at most \(n/p\).

A large prime \(p \geq n/L\) cannot be in surplus.

A medium prime \(\sqrt{n} {\lt} p ≤ n/L\) can be in deficit by at most \(M\).

A medium prime \(\sqrt{n} {\lt} p ≤ n/L\) can be in surplus by at most \(M\).

A small prime \(L {\lt} p \leq \sqrt{n}\) can be in deficit by at most \(M\log n\).

A small prime \(p \leq \sqrt{n}\) can be in surplus by at most \(M\log n\).

A tiny prime \(p \leq L\) can be in deficit by at most \(M\log n + ML\pi (n)\).

The total waste in this initial factorization is at most \(n \log \frac{1}{1-1/M}\).

The score of the initial factorization can be taken to be \(o(n)\).

The initial score is bounded by

The integral

is positive (and hence convergent - since a divergent integral is zero in Lean, by definition).

The conjugation symmetry of the interval integral. Let \(f : \mathbb {R} \to \mathbb {C}\) be a measurable function, and let \(a, b \in \mathbb {R}\). Then

Additive arithmetic function: satisfies \(f(mn) = f(m) + f(n)\) for coprime \(m\), \(n\).

Completely additive arithmetic function: satisfies \(f(mn) = f(m) + f(n)\) for all \(m, n \ne 0\).

A completely additive function is additive.

Define Complete Multiplicativity for an arithmetic function.

A function that is completely multiplicative is also multiplicative.

The Liouville function is completely multiplicative.

Let \(x{\gt}0\). Then the function \(f(s) = x^s/(s(s+1))\) is holomorphic on the half-plane \(\{ s\in \mathbb {C}:\Re (s){\gt}0\} \).

Let \(x{\gt}0\) and \(\sigma \in \mathbb {R}\). Then

is integrable.

For fixed \(\nu \) the function \(n\mapsto n^\nu \) is multiplicative.

For fixed \(s\) function \(n\mapsto \sigma ^R_s(n)\) is multiplicative.

The set \(\{ s\in \mathbb {C}\mid \Re (s){\gt}0 ∧ s \ne 1\} \) is path-connected.

Let \(0 {\lt} r {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)=1\), define \(L_f(z)=J_{B_f}(z)\) where \(J\) is from Theorem 5.3.3 and \(B_f\) is from Definition 5.3.6.

Let \(B{\gt}1\) and \(0 {\lt} r' {\lt} r {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)=1\) and \(|f(z)|\leq B\) for all \(|z|\leq R\), then for all \(|z|\leq r'\)

Let \(x\in \mathbb {R}\) and \(s \ne 0, -1\). Then

We have \(\sum _{n \leq x} \lambda (n) = o(x)\).

\(\mathrm{li}(x) = \int _0^x \frac{dt}{\log t}\) (in the principal value sense) and \(\mathrm{Li}(x) = \int _2^x \frac{dt}{\log t}\).

If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported and \(x \geq 1\), then

With the hypotheses as above, we have

as \(x \to \infty \).

If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported with \(f\) and \(\hat\psi \) non-negative, and \(0 {\lt} x\), then

Let \(a:\mathbb {R}\to \mathbb {C}\) be a function, and let \(\sigma {\gt}0\) be a real number. Suppose that, for all \(\sigma , \sigma '{\gt}0\), we have \(a(\sigma ')=a(\sigma )\), and that \(\lim _{\sigma \to \infty }a(\sigma )=0\). Then \(a(\sigma )=0\).

Let \(a:\mathbb {R}\to \mathbb {C}\) be a function, and let \(\sigma {\lt}-3/2\) be a real number. Suppose that, for all \(\sigma , \sigma '{\gt}0\), we have \(a(\sigma ')=a(\sigma )\), and that \(\lim _{\sigma \to -\infty }a(\sigma )=0\). Then \(a(\sigma )=0\).

Liouville function: \(\lambda (n) = (-1)^{\Omega (n)}\).

The Liouville function agrees with the Möbius function on square-free numbers.

If \(s {\gt} 1\) then \(\log \zeta (s) = - \log (s-1) + \Gamma '(1) + \gamma + (s-1) \int _1^\infty E_{2,\Lambda }(x) x^{-s}\ ds\).

If \(s {\gt} 1\) then \(\log \zeta (s) = - \log (s-1) + \Gamma '(1) + \gamma + (s-1) \int _1^\infty E_{2,\Lambda }(x) x^{-s}\ ds\).

If \(s {\gt} 1\) then \(\log \zeta (s) = (s-1) \int _1^\infty (\log \log x + \gamma + E_{2,\Lambda }(x)) x^{-s})\ dx\).

If \(s {\gt} 1\) then \((s-1) \int _1^\infty \log \log x \cdot x^{-s}\ dx = -\log (s-1) + \Gamma '(1)\).

If \(s {\gt} 1\) then \((s-1) \int _1^\infty \gamma \cdot x^{-s}\ dx = \gamma \).

One has \(\log \zeta (s) = - \log (s-1) + o(1)\) as \(s \to 1^+\).

For every \(0{\lt}n\) there exists a constant \(C{\gt}0\) (depending on \(n\)) such that

We have that, for \(\Re (s){\gt}1\),

If \(f\) is holomorphic in a neighborhood of \(p\), and there is a simple pole at \(p\), then \(f'/ f\) has a simple pole at \(p\) with residue \(-1\):

There exists a uniform constant \(C\) such that for \(z=\sigma +it\) one has

There is an \(A{\gt}0\) so that for \(1-A/\log ^9 |t| \le \sigma {\lt} 1+A/\log ^9 |t|\) and \(3 {\lt} |t|\),

There exist \(A, C {\gt} 0\) such that

whenever \(|t|{\gt}3\) and \(\sigma {\gt} 1 - A/\log |t|^9\).

Let \(t\in \mathbb {R}\) with \(|t|\geq 2\) and \(0 {\lt} r' {\lt} r {\lt} R' {\lt} R{\lt}1\). If \(f(z)=\zeta (z+3/2+it)\), then for all \(z\in \overline{\mathbb {D}_{r'}}\setminus \mathcal{K}_f(R')\) we have that

For constants \(A\), \(C\), \(n_1\), and \(n_2\) this is true if for all \(3{\lt}|t|\) one has

There is an \(A{\gt}0\) so that for all \(T{\gt}3\), the function \( \frac{\zeta '}{\zeta }(s) \) is holomorphic on \(\{ 1-A/\log ^9 T \le \Re s \le 2, |\Im s|\le T \} \setminus \{ 1\} \).

There is a \(\sigma _2 {\lt} 1\) so that the function

is holomorphic on \(\{ \sigma _2 \le \Re s \le 2, |\Im s| \le 3 \} \setminus \{ 1\} \).

There exists a constant \(F\in (0,1/42)\) such that for all \(t\in \mathbb {R}\) with \(|t|\geq 3\) one has

where the implied constant is uniform in \(\sigma \).

There exists a constant \(F\in (0,1/42)\) such that for all \(t\in \mathbb {R}\) with \(|t|\geq 3\) one has

where the implied constant is uniform in \(\sigma \).

Let \(0{\lt}r{\lt}R\). Let \(B:\overline{\mathbb {D}_{R}}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_{R}}\) with \(B(z)\neq 0\) for all \(z\in \overline{\mathbb {D}_{R}}\).Then there exists \(J_B:\mathbb {D}_R\to \mathbb {C}\) that is analytic on neighborhoods of points in \(\mathbb {D}_R\) such that

• \(J_B(0)=0\)

• \(J_B'(z)=B'(z)/B(z)\) for all \(z\in \overline{\mathbb {D}_r}\)

• \(\log |B(z)|-\log |B(0)|=\mathfrak {R}J_B(z)\) for all \(z\in \mathbb {D}_R\).

A wrapper of the above theorem that will be useful later on.

A LowerUIntegral of a function \(f\) comes from \(\sigma -i\infty \) up to \(\sigma -iT\), over to \(\sigma '-iT\), and back down to \(\sigma '-i\infty \).

If \(f\) is a multiplicative function, then so to is \(n\mapsto f(n)/n^s\).

The \(L\)-series of \(d_k\) equals \(\zeta (s)^k\) for \(\Re (s) {\gt} 1\).

The L-series for \(d_k\) is summable for \(\Re (s) {\gt} 1\).

The Dirichlet series of the Liouville function is \(\zeta (2s)/\zeta (s)\).

\(L(\text{pow}^R(\nu ), s) = \zeta (s - \nu )\) for \(\Re (s - \nu ) {\gt} 1\).

  This is IK (1.27).
  

\(\zeta (s)\zeta (s - \nu ) = \sum _{n=1}^{\infty } \sigma _\nu (n) n^{-s}\) for \(\Re (s) {\gt} 1\) and \(\Re (s - \nu ) {\gt} 1\).

The L-series of \(\tau \) equals the square of the Riemann zeta function for \(\Re (s) {\gt} 1\).

Euler totient series: \(\sum _{n=1}^{\infty } \varphi (n) n^{-s} = \zeta (s-1)/\zeta (s)\) for \(\Re (s) {\gt} 2\).

  This is IK (1.35). The L-series converges absolutely for Re(s) > 2
  (using φ(n) ≤ n to bound terms; the original hypothesis 1 < Re(s) is incorrect).
  

The \(L\)-series with coefficients given by \(\mu ^2(n)\) converges on the region \(1{\lt}\Re (s)\).

An L-series is convergent if the absolute value of each term is term wise less than a summable series.

We have

We define \(M := \int _2^\infty \frac{E_{1,p}(t)}{t \log ^2 t} \, dt + 1 - \log \log 2\).

One has \(M = \gamma + \sum _p \log (1-\frac{1}{p}) + \frac{1}{p}\).

Let \(f\) and \(g\) be functions from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\). Then we define the Mellin convolution of \(f\) and \(g\) to be the function \(f\ast g\) from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) defined by

Let \(f\) and \(g\) be functions from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {R}\) or \(\mathbb {C}\), for \(x\neq 0\),

Let \(f\) and \(g\) be functions from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) such that

is absolutely integrable on \([0,\infty )^2\). Then

The Mellin transform of \(1_{(0,1]}\) is

For any \(\epsilon {\gt}0\), the Mellin transform of \(\nu _\epsilon \) is

For any \(\epsilon {\gt}0\), we have

As \(\epsilon \to 0\), we have

The Mellin transform of \(\nu \) is

as \(|s|\to \infty \) with \(\sigma _1 \le \Re (s) \le 2\).

Fix \(\epsilon {\gt}0\). Then the Mellin transform of \(\widetilde{1_{\epsilon }}\) is

Given \(0{\lt}\sigma _1\le \sigma _2\), for any \(s\) such that \(\sigma _1\le \mathcal Re(s)\le \sigma _2\), we have

At \(s=1\), we have

We define \(E_{1,\Lambda }(x) := \sum _{d \leq x} \frac{\Lambda (d)}{d} - \log x\).

For any \(x \geq 1\), one has

For any \(x \geq 1\), one has

We define \(E_{1,p}(x) := \sum _{p \leq x} \frac{\log p}{p} - \log x\).

For any \(x \geq 1\), one has

and thus

where

For any \(x \geq 1\), one has

For any \(x \geq 1\), one has

For any \(x \geq 1\), one has

For any \(x \geq 1\), one has

For any \(x \geq 2\) and any \(f : {\mathbb N} \mapsto {\mathbb R}\), one has

One has \(M \geq -\frac{2 + E_1}{\log 2} + 1 - \log \log 2\).

One has \(M \leq \frac{\log 4 + 4}{\log 2} + 1 - \log \log 2\).

We define \(E_{2,\Lambda }(x) := \sum _{d \leq x} \frac{\Lambda (d)}{d \log d} - \log \log x - \gamma \).

For any \(x \geq 2\), one has

For any \(x \geq 2\), one has

For any \(x \geq 2\), one has

For any \(x \geq 2\), one has

For any \(x \geq 2\), one has

For any \(x \geq 2\), one has

For any \(x \geq 1\), one has

(NOTE: this identity is not actually needed in the proof of Mertens’ theorems, but may be worth recording nevertheless.)

For any \(x \geq 1\), one has

For any real \(x\), one has

For any \(x \geq 1\), one has

For any \(x \geq 1\), one has

We define \(E_3(x) := \sum _{p \leq x} \log (1 - \frac{1}{p}) + \log \log x + \gamma \).

For any \(x \geq 2\), one has

For any \(x \geq 2\), one has

I-K (1.33): \(\mu ^2(n) = \sum _{d^2|n} \mu (d)\).

We have that \(n\mapsto \mu ^2(n)/n^{s}\) is a multiplicative function.

For \(1{\lt}\Re (s)\) we have that

The naming convention here is designed to match

      riemannZeta_eulerProduct_hasProd
    

For \(1{\lt}\Re (s)\) we have that

The naming convention here is designed to match

      riemannZeta_eulerProduct_tprod
    

For \(p\) prime, we have that

We have \(\sum _{n \leq x} \mu (n) = o(x)\).

We have \(\sum _{n \leq x} \mu (n)/n = o(1)\).

We have \(|\sum _{n \leq x} \frac{\mu (n)}{n}| \leq 1\).

If a function \(f\) has a simple pole at a point \(p\) with residue \(A \neq 0\), then \(f\) is nonzero in a punctured neighborhood of \(p\).

Let \(0 {\lt} r {\lt} R{\lt}1\), and \(f:\mathbb {C}\to \mathbb {C}\) be analytic on \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). Then

If \(a,b\in \mathbb {R}\) with \(1{\lt}b{\lt}a\), then

Now let \(M,L\) be additional parameters with \(n {\gt} L^2\); we also need the minor variant \(\lfloor n/L \rfloor {\gt} \sqrt{n}\).

Let \(f, g\) be once differentiable functions from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) so that \(fg'\) and \(f'g\) are both integrable, and \(f\cdot g (x)\to 0\) as \(x\to 0^+,\infty \). Then

Let \(x{\gt}0\) and \(\sigma ',\sigma ''\in \mathbb {R}\). Then

goes to \(0\) as \(t\to \infty \).

\(\pi (x)\) is the number of primes less than or equal to \(x\).

\(\pi ^*(x) = \sum _{k \geq 1} \pi (x^{1/k}) / k\).

One has

as \(x \to \infty \).

There exists a function \(c(x)\) such that \(c(x) = o(1)\) as \(x \to \infty \) and

for all \(x\) large enough.

One has

as \(n \to \infty \).

We have \(p_{n+1} - p_n = o(p_n)\) as \(n \to \infty \).

Let \(m\) and \(n\) be coprime natural numbers and let \(k\) be fixed. The divisors \(x\) of \(mn\) with \(x^k \mid mn\) are exactly the products \(ab\) where \(a \mid m\), \(a^k \mid m\), \(b \mid n\), and \(b^k \mid n\).

Let \(m\) and \(n\) be coprime natural numbers and let \(k\) be fixed. The map \((a,b) \mapsto ab\) is injective on the product of \(\{ a \mid m : a^k \mid m\} \) and \(\{ b \mid n : b^k \mid n\} \).

If \(f\) is an additive function, then so to is \(n\mapsto k^{f(n)}\) for any fixed \(k\).

If \(f\) is a multiplicative function, then so to is \(n\mapsto f(n)^k\) for any \(k\in \mathbb {N}\).

Arithmetic function with complex parameter \(\nu \). Evaluates as \(n\mapsto n^{\nu }\) for \(n\neq 0\) and \(0\) at \(n=0\).

If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable then

for all \(u \in \mathbb {R}\). where \(C\) is an absolute constant.

If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable and of bounded variation, then

for all non-zero \(u \in \mathbb {R}\).

If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable, absolutely continuous, and \(\psi '\) is of bounded variation, then

for all non-zero \(u \in \mathbb {R}\).

For every \(\varepsilon {\gt}0\), there is a prime between \(x\) and \((1+\varepsilon )x\) for all sufficiently large \(x\).

For all \(n \geq 2\), one has

We have

For all \(0{\lt}c\) and \(0{\lt}x\) we have that

A Rectangle’s border, given corners \(z\) and \(w\) is the union of the four sides.

A RectangleIntegral of a function \(f\) is one over a rectangle determined by \(z\) and \(w\) in \(\mathbb {C}\). We will sometimes denote it by \(\int _{z}^{w} f\). (There is also a primed version, which is \(1/(2\pi i)\) times the original.)

Let \(\sigma ,\sigma ' \in \mathbb {R}\), and \(f : \mathbb {C} \to \mathbb {C}\) such that the vertical integrals \(\int _{(\sigma )}f(s)ds\) and \(\int _{(\sigma ')}f(s)ds\) exist and the horizontal integral \(\int _{(\sigma )}^{\sigma '}f(x + yi)dx\) vanishes as \(y \to -\infty \). Then the limit of rectangle integrals

as \(U\to \infty \) is the “LowerUIntegral” of \(f\).

Let \(\sigma ,\sigma ' \in \mathbb {R}\), and \(f : \mathbb {C} \to \mathbb {C}\) such that the vertical integrals \(\int _{(\sigma )}f(s)ds\) and \(\int _{(\sigma ')}f(s)ds\) exist and the horizontal integral \(\int _{(\sigma )}^{\sigma '}f(x + yi)dx\) vanishes as \(y \to \pm \infty \). Then the limit of rectangle integrals

as \(U\to \infty \) is the “UpperUIntegral” of \(f\).

Let \(\sigma ,\sigma ' \in \mathbb {R}\), and \(f : \mathbb {C} \to \mathbb {C}\) such that the vertical integrals \(\int _{(\sigma )}f(s)ds\) and \(\int _{(\sigma ')}f(s)ds\) exist and the horizontal integral \(\int _{(\sigma )}^{\sigma '}f(x + yi)dx\) vanishes as \(y \to \pm \infty \). Then the limit of rectangle integrals

If \(f\) is holomorphic on a rectangle \(z\) and \(w\) except at a point \(p\), then the integral of \(f\) over the rectangle with corners \(z\) and \(w\) is the same as the integral of \(f\) over a small square centered at \(p\).

Let \(x{\gt}0\). Then for all sufficiently small \(c{\gt}0\), we have that

Let \(x{\gt}0\). Then for all sufficiently small \(c{\gt}0\), we have that

If \(f\) has a simple pole at \(p\) with residue \(A\), and \(g\) is holomorphic near \(p\), then the residue of \(f \cdot g\) at \(p\) is \(A \cdot g(p)\). That is, we assume that

near \(p\), and that \(g\) is holomorphic near \(p\). Then

If a function \(f\) is holomorphic in a neighborhood of \(p\) and \(\lim _{s\to p} (s-p)f(s) = A\), then \(f(s) = \frac{A}{s-p} + O(1)\) near \(p\).

For \(x{\gt}1\) (of course \(x{\gt}0\) would suffice) and \(\sigma {\gt}0\), we have

For \(x{\gt}1\), we have

The rectangle (square) integral of \(f(s) = 1/s\) with corners \(-1-i\) and \(1+i\) is equal to \(2\pi i\).

Suppose that \(f\) is a holomorphic function on a rectangle, except for a simple pole at \(p\). By the latter, we mean that there is a function \(g\) holomorphic on the rectangle such that, \(f = g + A/(s-p)\) for some \(A\in \mathbb {C}\). Then the integral of \(f\) over the rectangle is \(A\).

For any natural \(N\ge 1\), we define

Let

The log derivative of the Riemann zeta function \(\zeta (s)\) has a simple pole at \(s=1\) with residue \(-1\): \(-\frac{\zeta '(s)}{\zeta (s)} - \frac{1}{s-1} = O(1)\).

The Riemann zeta function \(\zeta (s)\) has a simple pole at \(s=1\) with residue \(1\). In particular, the function \(\zeta (s) - \frac{1}{s-1}\) is bounded in a neighborhood of \(s=1\).

The previous corollary also holds for functions \(\psi \) that are assumed to be in the Schwartz class, as opposed to being \(C^2\) and compactly supported.

The score of a factorization (relative to a cutoff parameter \(L\)) is equal to its waste, plus \(\log p\) for every surplus prime \(p\), \(\log (n/p)\) for every deficit prime above \(L\), \(\log L\) for every deficit prime below \(L\) and an additional \(\log n\) if one is not in total balance.

If one is in total balance, then the score is equal to the waste.

If there is a prime \(p\) in surplus, one can remove it without increasing the score.

If there is a prime \(p\) in deficit larger than \(L\), one can remove it without increasing the score.

If there is a prime \(p\) in deficit less than \(L\), one can remove it without increasing the score.

One can bring any factorization into balance without increasing the score.

If \(\psi : \mathbb {R}\to \mathbb {C}\) is absolutely integrable and \(x {\gt} 0\), then for any \(\sigma {\gt}1\)

Let \(R{\gt}0\) and \(f:\mathbb {C}\to \mathbb {C}\). Define the set of zeros \(\mathcal{K}_f(R)=\{ \rho \in \mathbb {C}:|\rho |\leq R,\, f(\rho )=0\} \).

There exists \(C{\gt}0\) such that for all \(\delta \in (0,1)\) and \(t\in \mathbb {R}\) with \(|t|\geq 2\); if \(\zeta (\rho )=0\) with \(\rho =\sigma +it\), then

For all \(\delta \in (0,1)\) and \(t\in \mathbb {R}\) with \(|t|\geq 2\) we have

For all \(\delta \in (0,1)\) we have

Divisor power sum with complex exponent.

We have that \(\sigma ^R_s(n)=\sum _{d\mid n}d^s.\)

For a prime power, we have that \(\sigma ^R_s(p^i)=\sum _{j=0}^ip^{js}\).

We have that

We have that \(\sigma ^R_s(n)=\sum _{d\mid n}(n/d)^s\).

\(\sigma ^R(\nu ) = \zeta * \text{pow}^R(\nu )\), where \(\zeta \) is the constant function \(1\).

For natural exponents, \(\sigma ^R\) agrees with \(\sigma \).

A casting lemma for \(\sigma ^R\).

We have that \(\sigma ^R_s(1)=1\).

Same as the previous lemma, but with a different casting structure.

Same as the previous lemma, but with a different casting structure.

Same as the previous lemma, but with a different casting structure.

Same as the previous lemma, but with a different casting structure.

If \(I\) is a closed interval contained in an open interval \(J\), then there exists a smooth function \(\Psi : \mathbb {R}\to \mathbb {R}\) with \(1_I \leq \Psi \leq 1_J\).

Let \(\epsilon {\gt}0\). Then we define the smooth function \(\widetilde{1_{\epsilon }}\) from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) by

Fix a nonnegative, continuously differentiable function \(F\) on \(\mathbb {R}\) with support in \([1/2,2]\). Then for any \(\epsilon {\gt}0\), the function \(x \mapsto \int _{(0,\infty )} x^{1+it} \widetilde{1_{\epsilon }}(x) dx\) is continuous at any \(y{\gt}0\).

If \(\nu \) is nonnegative and has mass one, then \(\widetilde{1_{\epsilon }}(x)\le 1\), \(\forall x{\gt}0\).

If \(\nu \) is nonnegative, then \(\widetilde{1_{\epsilon }}(x)\) is nonnegative.

Fix \(0{\lt}\epsilon {\lt}1\). There is an absolute constant \(c{\gt}0\) so that: if \(x\geq (1+c\epsilon )\), then

Fix \(\epsilon {\gt}0\). There is an absolute constant \(c{\gt}0\) so that: If \(0 {\lt} x \leq (1-c\epsilon )\), then

Fix \(\epsilon {\gt}0\), and a bumpfunction supported in \([1/2,2]\). Then we define the smoothed Chebyshev function \(\psi _{\epsilon }\) from \(\mathbb {R}_{{\gt}0}\) to \(\mathbb {C}\) by

where we’ll take \(\sigma = 1 + 1 / \log X\).

We have that

We have that

Fix a nonnegative, continuously differentiable function \(F\) on \(\mathbb {R}\) with support in \([1/2,2]\), and total mass one, \(\int _{(0,\infty )} F(x)/x dx = 1\). Then for any \(\epsilon {\gt}0\), and \(\sigma \in (1, 2]\), the function

is integrable on \(\mathbb {R}\).

Fix a nonnegative, continuously differentiable function \(F\) on \(\mathbb {R}\) with support in \([1/2,2]\), and total mass one, \(\int _{(0,\infty )} F(x)/x dx = 1\). Then for any \(\epsilon {\gt}0\) and \(\sigma \in (1,2]\), the function \(x \mapsto \sum _{n=1}^\infty \frac{\Lambda (n)}{n^{\sigma +it}} \mathcal{M}(\widetilde{1_{\epsilon }})(\sigma +it) x^{\sigma +it}\) is equal to \(\sum _{n=1}^\infty \int _{(0,\infty )} \frac{\Lambda (n)}{n^{\sigma +it}} \mathcal{M}(\widetilde{1_{\epsilon }})(\sigma +it) x^{\sigma +it}\).

We have that

The integrand

is integrable on the contour \(\sigma _0 + t i\) for \(t \in \mathbb {R}\) and \(\sigma _0 {\gt} 1\).

We have that

There exists a smooth (once differentiable would be enough), nonnegative “bumpfunction” \(\nu \), supported in \([1/2,2]\) with total mass one:

We have

If \(f\) is a multiplicative arithmetic function, then for coprime, nonzero \(a\) and \(b\), we have that \(\sum _{d | ab} f(d) = (\sum _{d | a} f(d)) \cdot (\sum _{d | b} f(d))\).

  This has been upstreamed #36495.
  

Let \(a {\lt} b\), and let \(\phi \) be continuously differentiable on \([a, b]\). Then

If \(g\) is a multiplicative arithmetic function, then for any \(n \neq 0\), \(\sum _{d | n} \mu (d) \cdot g(d) = \prod _{p | n} (1 - g(p))\).

  Upstream issue has been created #1240.
  

The function \(n \mapsto \sum _{d^2|n} \mu (d)\).

A simple helper lemma for the above definition.

Applied at prime powers, sum-moebius-sq-divisors coincides with \(\mu ^2\).

The function \(n \mapsto \sum _{d^2|n} \mu (d)\) is multiplicative.

For all \(\delta \in (0,1)\) and \(t\in \mathbb {R}\) with \(|t|\geq 2\) we have

For all \(t\in \mathbb {R}\) with \(|t|\geq 2\) and \(z=\sigma +it\) where \(1-\delta _t/3\leq \sigma \leq 3/2\), we have that

Shorthand for the sum of a function \(f:\mathbb {N}\to \mathbb {C}\) at prime powers for a prime \(p\).

Helper lemma for sumOnPrimePows that rewrites the shorthand as its sum.

Let \(0 {\lt} R\). Let \(f:\overline{\mathbb {D}_R}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_R}\). Define the functon \(I_f:\mathbb {D}_R\to \mathbb {C}\) by

Let \(x{\gt}1\). Then

Let \(x{\gt}0\) and \(x{\lt}1\). Then

Let \(x{\gt}0\) and \(\sigma ',\sigma ''\in \mathbb {R}\). Then

goes to \(0\) as \(t\to -\infty \).

For any \(p{\gt}0\) and \(t\in \mathbb {R}\),

There is a constant \(c{\gt}0\), so that \(\zeta (s)\) does not vanish in the region \(\sigma {\gt}1-\frac{c}{\log t}\), and moreover,

there.

The function \(n\mapsto 2^{\omega (n)}\) is multiplicative.

We have the inequality \(2^{\omega (n)}\leq \sigma _0(n)\) when \(n\neq 0\).

We have that \(n\mapsto 2^{\omega (n)}/n^{s}\) is a multiplicative function.

For \(1{\lt}\Re (s)\) we have that

The naming convention here is designed to match

      riemannZeta_eulerProduct_hasProd
    

For \(1{\lt}\Re (s)\) we have that

The naming convention here is designed to match

      riemannZeta_eulerProduct_tprod
    

For \(1{\lt}\Re (s)\) and \(p\) prime, we have that

If \(a\) and \(b\) are coprime, then any divisor \(d\) of \(ab\) can be uniquely expressed as a product of a divisor of \(a\) and a divisor of \(b\).

  This has been upstreamed #36495.
  

An UpperUIntegral of a function \(f\) comes from \(\sigma +i\infty \) down to \(\sigma +iT\), over to \(\sigma '+iT\), and back up to \(\sigma '+i\infty \).

Let \(f\) be a function from \(\mathbb {C}\) to \(\mathbb {C}\), and let \(\sigma \) be a real number. Then we define

Let \(x{\gt}0\) and \(\sigma {\gt}1\). Then

Let \(x{\gt}1\) and \(\sigma {\lt}-3/2\). Then

The series

converges when \(1{\lt}\mathfrak {R}(s)\).

The waste of a factorizations \(a_1 \dots a_t\) is defined as \(\sum _i \log (n / a_i)\).

The waste of a factorization is equal to \(t \log n - \log n!\), where \(t\) is the number of elements.

We have

If \(q ≥ 1\) and \(a\) is coprime to \(q\), we have

If \(q ≥ 1\) and \(a\) is coprime to \(q\), the Dirichlet series \(\sum _{n \leq x: n = a\ (q)} \frac{\Lambda (n)}{n^s}\) converges for \(\mathrm{Re}(s) {\gt} 1\) to \(\frac{1}{\varphi (q)} \frac{1}{s-1} + G(s)\) where \(G\) has a continuous extension to \(\mathrm{Re}(s)=1\).

If \(q ≥ 1\) and \(a\) is coprime to \(q\), and \(\mathrm{Re} s {\gt} 1\), we have

We have

We have

For any closed interval \(I \subset (0,+\infty )\), we have

If \(\Psi : (0,\infty ) \to \mathbb {C}\) is smooth and compactly supported away from the origin, then,

as \(x \to \infty \).

Let \(f:\mathbb {C}\to \mathbb {C}\) and \(\rho \in \mathbb {C}\). Then there exists \(h_\rho \) such that

In LEAN, this corresponds exactly with (-.analyticOrderAt-ne-top.mp -).choose, but this serves as a wrapper of that with the necessary conditions.

Let \(f:\mathbb {C}\to \mathbb {C}\) be analytic on \(\overline{\mathbb {D}_1}\) with \(f(0)\neq 0\). For all \(\rho \in \mathcal{K}_f(R)\) with \(R{\lt}1\) there exists \(h_\rho (z)\) such that \(h_\rho (z)\) is analytic at \(\rho \), \(h_\rho (\rho )\neq 0\), and \(f(z)=(z-\rho )^{m_f(\rho )}\, h_\rho (z)\).

There exists a constant \(0 {\lt} E{\lt}1/14\) such that for all \(\rho =\sigma +it\) with \(\zeta (\rho )=0\) and \(|t|\geq 2\), one has

Let \(f:\mathbb {C}\to \mathbb {C}\). We define \(m_f(\rho )\) as the order of the zero \(\rho \) w.r.t \(f\).

Let \(0{\lt} r {\lt} R{\lt}1\). If \(f:\mathbb {C}\to \mathbb {C}\) is a function analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\) with \(f(0)=1\) and \(|f(z)|\leq B\) for \(|z|\leq R\), then

If the limit of \(0\) is \(L_1 - L_2\), then \(L_1 = L_2\).

Let \(\mathcal{Z}_t=\{ \rho \in \mathbb {C}:\zeta (\rho )=0,\, |\rho -(3/2+it)|\leq 3/4\} \).

For all \(|t|\geq 2\), \(\mathcal{Z}(t)\) contains a finite number of points.

For \(\Re (s){\gt}0\), \(s\ne 1\), and for any \(N\),

We have that

where \(\zeta _0(1,s)\) comes from Definition 4.4.1.

    An expression for `ζ`, in IK (1.32).
  

For any \(A{\gt}0\) sufficiently small, there is a constant \(C{\gt}0\) so that whenever \(1- A / \log t \le \sigma _1 {\lt} \sigma _2\le 2\) and \(3 {\lt} |t|\), we have that:

For any \(t\ne 0\) (so we don’t pass through the pole), and \(\sigma _1 {\lt} \sigma _2\),

Baby Rankin-Selberg: \(\zeta (s)\sum _{n=1}^{\infty }\tau (n^2)n^{-s} = \sum _{n=1}^{\infty }\tau (n)^2 n^{-s}\) for \(\Re (s) {\gt} 1\).

  Precursor to IK (1.30).
  

The Dirichlet convolution of \(\zeta \) with itself is \(\tau \) (the divisor count function).

Ramanujan formula: \(\zeta (s)\zeta (s-\alpha )\zeta (s-\beta )\zeta (s-\alpha -\beta )=\zeta (2s-\alpha -\beta ) \sum _{n=1}^{\infty } \sigma _\alpha (n)\sigma _\beta (n)n^{-s}\) for \(\Re (s) {\gt} 1\), \(\Re (s-\alpha ) {\gt} 1\), \(\Re (s-\beta ) {\gt} 1\), and \(\Re (s-\alpha -\beta ) {\gt} 1\).

  This is IK (1.28).
  

Corollary: \(\zeta (s)^4 = \zeta (2s) \sum _{n=1}^{\infty } \tau (n)^2 n^{-s}\) for \(\Re (s) {\gt} 1\).

  This is IK (1.29).
  

Zeta cubed: \(\zeta (s)^3 = \zeta (2s) \sum _{n=1}^{\infty }\tau (n^2) n^{-s}\) for \(\Re (s) {\gt} 1\).

  This is IK (1.30).
  

Zeta cubed: \(\zeta (s)^3 = \zeta (2s) \sum _{n=1}^{\infty }\tau (n^2) n^{-s}\).

  This is IK (1.30).
  

symmetric square \(L\)-function for \(\zeta ^2\):

for \(\Re (s) {\gt} 1\).

  Alternative expression for `ζ^3`, in IK between (1.30) and (1.31).
  

for \(\Re (s) {\gt} 1\).

    An expression for `ζ^2`, in IK (1.31).
  

We have that

for all \(s\in S\) with \(S=\{ s\in \mathbb {C}:0{\lt}\mathfrak {R}s,\, s\neq 1\} \).

For any \(N\ge 1\) and \(s = \sigma + tI \in \mathbb {C}\), \(\sigma =\in (0,2], 2 {\lt} |t|\),

For any \(0 {\lt} a {\lt} b\) and \(s \in \mathbb {C}\) with \(\sigma =\Re (s){\gt}0\),

For any \(N\ge 1\) and \(s = \sigma + tI \in \mathbb {C}\), \(\sigma {\gt} 0\),

For any \(N\ge 1\) and \(s = \sigma + tI \in \mathbb {C}\), \(\sigma =\in (0,2], 2 {\lt} |t|\),

Given \(n ≤ t\) and \(\sigma \) with \(1-A/\log t \le \sigma \), we have that

For all \(\epsilon {\gt} 0\) sufficiently close to \(0\), the rectangle integral over \([1-\delta ,2] \times _{ℂ} [-T,T]\) of the integrand in \(\psi _{\epsilon }\) is

where the implicit constant is independent of \(X\).

For any \(s = \sigma + tI \in \mathbb {C}\), \(1/2 \le \sigma \le 2, 3 {\lt} |t|\), there is an \(A{\gt}0\) so that for \(1-A/\log t \le \sigma \), we have

For all \(t\in \mathbb {R}\) one has

For any \(A{\gt}0\) sufficiently small, there is a constant \(C{\gt}0\) so that whenever \(1- A / \log ^9 |t| \le \sigma {\lt} 1+A/\log ^9 |t|\) and \(3 {\lt} |t|\), we have that:

For all \(\sigma {\gt}1\),

For \(\sigma {\gt}1\) (and \(\sigma \le 2\)),

as \(|t|\to \infty \).

For any \(A{\gt}0\) sufficiently small, there is a constant \(C{\gt}0\) so that whenever \(1- A / \log ^9 |t| \le \sigma {\lt} 1\) and \(3 {\lt} |t|\), we have that:

There exists a \(c{\gt}0\) such that for all \(1 {\lt} \sigma {\lt}= 2\) and \(3 {\lt} |t|\),

There exists a \(c{\gt}0\) such that for all \(1 {\lt} \sigma ≤ 2\),

As \(\sigma \to 1^+\),

For any \(T{\gt}0\), there is a constant \(\sigma {\lt}1\) so that

for all \(|t| \leq T\) and \(\sigma ' \ge \sigma \).

The zeta function does not vanish on the 1-line.

For all \(|t|\geq 2\), if \(f(z)=\zeta (z+3/2+it)\), then \(f(z)\) has a finite number of zeros in \(\overline{\mathbb {D}_1}\).

Let \(0 {\lt} a {\lt} b\) be natural numbers and \(s\in \mathbb {C}\) with \(s \ne 1\) and \(s \ne 0\). Then

Let \(N\) be a natural number and \(s\in \mathbb {C}\), \(\Re (s){\gt}1\). Then

For any \(s = \sigma + tI \in \mathbb {C}\), \(1/2 \le \sigma \le 2, 3 {\lt} |t|\) and any \(0 {\lt} A {\lt} 1\) sufficiently small, and \(1-A/\log |t| \le \sigma \), we have

There is an \(A{\gt}0\) so that for \(1-A/\log ^9 |t| \le \sigma {\lt} 1\) and \(3 {\lt} |t|\),